Graph coloring and semidefinite rank

Abstract

This paper considers the interplay between semidefinite programming, matrix rank, and graph coloring. Karger et al. (J ACM 45(2):246–265, 1998) give a vector program in which a coloring of a graph can be encoded as a semidefinite matrix of low rank. By complementary slackness conditions of semidefinite programming, if an optimal dual solution has high rank, any optimal primal solution must have low rank. We attempt to characterize graphs for which we can show that the corresponding dual optimal solution must have rank high enough that the primal solution encodes a coloring. In the case of the original Karger, Motwani, and Sudan vector program, we show that any graph which is a k-tree has sufficiently high dual rank, and we can extract the coloring from the corresponding low-rank primal solution. We can also show that if a graph is not uniquely colorable, then no sufficiently high rank dual optimal solution can exist. This allows us to completely characterize the planar graphs for which dual optimal solutions have sufficiently high dual rank, since it is known that the uniquely colorable planar graphs are precisely the planar 3-trees. We then modify the semidefinite program to have an objective function with costs, and explore when we can create an objective function such that the optimal dual solution has sufficiently high rank. We show that it is always possible to construct such an objective function given the graph coloring. The construction of the objective function gives rise to heuristics for 4-coloring planar graphs. We enumerated all maximal planar graphs with an induced $K_4$ of up to 14 vertices; the heuristics successfully found a 4-coloring for 99.75% of them. Our research was motivated by trying to use semidefinite programming to prove the four-color theorem, which states that every planar graph can be colored with four colors. There is an intriguing connection of the Karger–Motwani–Sudan semidefinite program with the Colin de Verdière graph invariant (J Combin. Theory Ser B 50:11-21, 1990) (and a corresponding conjecture of Colin de Verdière), in which matrices that have some similarities to the dual feasible matrices of the semidefinite program must have high rank in the case that graphs are of a certain type; for instance, planar graphs have rank that would imply that the primal solution of the semidefinite program encodes a 4-coloring.